Quantum Chemistry/Example 26

The square of the angular momentum of a hydrogen atom is measured to be $L^{2}$$=20\hbar ^{2}$.  What are the possible values of the z-component of the orbital angular momentum, $L_{z}$, that could be measured for this atom?

Solution

The eigenvalues of the square of the angular momentum operator (${\hat {L}}$²) for a quantum mechanical system, such as an electron in a hydrogen atom, are given by:

${\hat {L}}^{2}Y_{l}^{m}(\theta ,\phi )=h^{2}l(l+1)Y_{l}^{m}(\theta ,\phi )$

where $Y_{l}^{m}(\theta ,\phi )$ are the spherical harmonics, which are eigenfunctions of ${\hat {L}}$², $l$ is the orbital quantum number, and $m$ is the magnetic quantum number. The given value for $L^{2}$ = 20 ħ2 , so we set up the equation:

$\hbar ^{2}l(l+1)=20\hbar ^{2}$

Dividing by $\hbar ^{2}$ and simplifying, we get:

$l(l+1)=20$

$l^{2}+l-20=0$

In this quadratic equation, $l$ can be factored to get:

$(l-4)(l+5)=0$

$l=4$

Since $l$ must be a non-negative integer. The magnetic quantum number $m$ can take on any integer value from $-l$ to $l$, thus for $l=4$, $m$ can be:

$m=-4,-3,-2,-1,0,1,2,3,4$

The z-component of the angular momentum, $L_{z}$, is quantized in units of $\hbar ^{2}$and given by:

$L_{z}=m\hbar$

$L_{z}=-4\hbar ,-3\hbar ,-2\hbar ,-1\hbar ,0\hbar ,1\hbar ,2\hbar ,3\hbar ,4\hbar$.

Therefore, the possible values of the z-component of the orbital angular momentum, $L_{z}$, that could be measured for the atom with a given $L^{2}$$=20\hbar ^{2}$ are $L_{z}=-4\hbar ,-3\hbar ,-2\hbar ,-1\hbar ,0\hbar ,1\hbar ,2\hbar ,3\hbar ,4\hbar$.